Abstract and Applied Analysis

Volume 2013, Article ID 138068, 8 pages

http://dx.doi.org/10.1155/2013/138068

## Asymptotic Behavior of Solutions to Abstract Stochastic Fractional Partial Integrodifferential Equations

^{1}Department of Mathematics and Information Engineering, Sanming University, Sanming, Fujian 365004, China^{2}Department of Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, China^{3}Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela, 15782 Santiago de Compostela, Spain^{4}Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 28 June 2013; Accepted 1 December 2013

Academic Editor: Dumitru Baleanu

Copyright © 2013 Zhi-Han Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The existence of asymptotically almost automorphic mild solutions to an abstract stochastic fractional partial integrodifferential equation is considered. The main tools are some suitable composition results for asymptotically almost automorphic processes, the theory of sectorial linear operators, and classical fixed point theorems. An example is also given to illustrate the main theorems.

#### 1. Introduction

This paper is mainly concerned with the existence and uniqueness of square-mean asymptotically almost automorphic mild solutions to the following stochastic fractional partial integrodifferential equation in the form where , is a linear densely defined operator of sectorial type on a Hilbert space , is a two-sided standard one-dimensional Brownian motion defined on the filtered probability space , where , and is an -adapted, -valued random variable independent of the Wiener process . Here and are appropriate functions to be specified later. The convolution integral in (1) is understood in the Riemann-Liouville fractional integral (see, e.g., [1, 2]). We notice that fractional order can be complex in viewpoint of pure mathematics and there is much interest in developing the theoretical analysis and numerical methods to fractional equations, because they have recently proven to be valuable in various fields of science and engineering (see, e.g., [3–11] and references therein).

The concept of asymptotically almost automorphic functions was firstly introduced by N’Guérékata in [12]. Since then these functions have become of great interest to several mathematicians and gained lots of developments and applications, we refer the reader to [13–16] and the references listed therein.

Recently, the existence of almost automorphic and pseudo almost automorphic solutions to some stochastic differential equations has been considered in many publications such as [17–27] and the references therein. In a very recent paper [28], the authors introduced a new notation of square-mean asymptotically almost automorphic stochastic processes including a composition theorem. However, to the best of our knowledge, the existence of square-mean asymptotically almost automorphic mild solutions to the problem (1) is an untreated topic. Therefore, motivated by the works [16, 28], the main purpose of this paper is to investigate the existence and uniqueness of square-mean asymptotically almost automorphic mild solutions to the problem (1). Then, we present an example as an application of our main results.

The rest of this paper is organized as follows. In Section 2, we recall some basic definitions and facts which will be used throughout this paper. In Section 3, we prove some existence results of square-mean asymptotically almost automorphic mild solutions to the problem (1). Finally, we give an example as an application of our abstract results.

#### 2. Preliminaries

In this section, we introduce some basic definitions, notations, and preliminary facts which will be used in the sequel. For more details on this section, we refer the reader to [28–30].

Throughout the paper, stands for a real separable Hilbert space. denotes a complete probability space, and stands for the space of all -valued random variables such that Note that is a Hilbert space equipped with the norm We denote by the collection of all bounded continuous stochastic processes from into such that . It is then easy to check that is a Banach space when it is endowed with the norm . Similarly, stands for the space of the continuous stochastic processes such that uniformly for , where is any bounded subset. Additionally, will be a two-sided standard one-dimensional Brownian motion defined on the filtered probability space , where .

##### 2.1. Sectorial Linear Operators

A closed and linear operator is said to be sectorial of type and angle if there exist , and such that its resolvent exists outside the sector and , .

*Definition 1 (see [2]). *Let be a closed and linear operator with domain defined on a Banach space . We call the generator of a solution operator if there exist and a strongly continuous function such that and , . In this case, is called the solution operator generated by .

We note that if is sectorial of type with , then is the generator of a solution operator given by , where is a suitable path lying outside the sector . Recently, Cuesta in [1] proved that if is a sectorial operator of type for some and , then there exists a constant such that

*Remark 2. *Note that is, in fact, integrable. For more details on the solution family and related issues, we refer the reader to [31–33].

##### 2.2. Square-Mean Asymptotically Almost Automorphic Processes

We recall some basic facts for a symptotically almost automorphic processes which will be used in the sequel.

*Definition 3 (see [22]). *A stochastic process is said to be stochastically continuous if

*Definition 4 (see [17]). *A stochastically continuous stochastic process is said to be square-mean almost automorphic if, for every sequence of real numbers , there exist a subsequence and a stochastic process such that
hold for each . The collection of all square-mean almost automorphic stochastic processes is denoted by .

*Definition 5 (see [17]). *A function , , which is jointly continuous, is said to be square-mean almost automorphic if is square-mean almost automorphic in uniformly for all , where is any bounded subset of . That is to say, for every sequence of real numbers , there exists a subsequence and a function such that
for each and each . Denote by the set of all such functions.

Lemma 6 (see [22]). * is a Banach space equipped with the norm
**
for .*

Lemma 7 (see [17]). *Let , be square-mean almost automorphic, and assume that is uniformly continuous on each bounded subset uniformly for ; that is, for all , there exists such that and imply that for all . Then for any square-mean almost automorphic process , the stochastic process given by is square-mean almost automorphic.*

*Definition 8 (see [25]). *A stochastically continuous process is said to be square-mean asymptotically almost automorphic if it can be decomposed as , where and . Denote by the collection of all the square-mean asymptotically almost automorphic processes .

*Definition 9 (see [28]). *A function , , which is jointly continuous, is said to be square-mean asymptotically almost automorphic if it can be decomposed as , where and . Denote by the set of all such functions.

Lemma 10 (see [28]). *If , , and are all square-mean asymptotically almost automorphic stochastic processes, then the following hold true: * (I)* is square-mean asymptotically almost automorphic;* (II)* is square-mean asymptotically almost automorphic for any scalar ;* (III)*there exists a constant such that .*

Lemma 11 (see [28]). *Suppose that admits a decomposition , where and . Then .*

Corollary 12 (see [28]). *The decomposition of a square-mean asymptotically almost automorphic process is unique.*

Lemma 13 (see [28]). * is a Banach space when it is equipped with the norm
**
where with , .*

Lemma 14 (see [28]). * is a Banach space with the norm
*

*Remark 15 (see [28]). *In view of the previous lemmas it is clear that the two norms are equivalent in .

Lemma 16 (see [28]). *Let and let be uniformly continuous in any bounded subset uniformly for . Then is uniformly continuous in any bounded subset uniformly for .*

Lemma 17 (see [28]). *Let and suppose that is uniformly continuous in any bounded subset uniformly for . If , then .*

We now give the following concept of mild solution of (1).

*Definition 18. *Let be an integrable solution operator on with generator . An -adapted stochastic process is called a mild solution of the problem (1) if is -measurable and satisfies the corresponding stochastic integral equation:

#### 3. Main Results

In this section, we establish the existence of square-mean asymptotically almost automorphic mild solutions to the problem (1). For that, we need the following technical results.

First, we list the following basic assumptions.(H1)The operator is a sectorial operator of type for some and , and then there exists such that where is the solution operator generated by .(H2)The function and there exists a continuous and nondecreasing function such that for each and for all , , for all .(H3)The function and there exists a continuous and nondecreasing function such that for each and for all , , for all .(H4)We have where and .(H5)The operator is a sectorial operator of type with , and there exists such that where is the solution operator generated by .

Lemma 19. *Suppose that assumption (H1) holds and let . If is the function defined by
**
then .*

*Proof. *Since , we have by definition that , where ; and . Then
where and .

First we prove that . Let be an arbitrary sequence of real numbers. Since , there exists a subsequence of such that for a certain stochastic process
hold for each . Now, let for each . Note that is also a Brownian motion and has the same distribution as . Moreover, if we let , then by making a change of variables to get (see the equation (10.6.6) in [34])
and, hence, using the Ito’s isometry property of stochastic integral, we have the following estimations
Then by (20), we obtain that for each . In a similar way, we can show that for each . Thus we conclude that .

Next, let us show that . Since and is integrable in , for any sufficiently small , there exists a constant such that and for all . Then, for all , we obtain
This inequality proves the assertion since is arbitrary. Recalling that for all , we get . The proof is completed.

It is easy to see that, by arguments similar to those in the proof of Lemma 19, we have the following result.

Lemma 20. *Suppose that assumption (H5) holds and let . If is the function defined by
**
then .*

Now, we are ready to establish our main results.

Theorem 21. *Assume that (H1)–(H4) hold. Then there exists such that for each there exists a unique square-mean asymptotically almost automorphic mild solution of the problem (1) on such that .*

*Proof. *We define a nonlinear operator by

First we prove that . Given , from the properties of , , and , we infer that is well defined and continuous. Since is bounded, we can choose a bounded subset of such that for all . It follows from (H2) and (H3) that both and are uniformly continuous on the bounded subset uniformly for . Moreover, from Lemmas 17 and 19 and taking into account (H1), it follows that .

Now, by (H4), there exists a constant such that
Let . We affirm that the assertion holds for . In fact, let . Define the space . Then is a closed subspace of . We claim that . If and , we get
which from (26) implies that for all , and so that .

Next, to complete the proof, we need to show that is a contraction from into . By (26), we know that
That is,
Then, one has
For any and , we have
Thus, we get
It follows from (30) that is a contraction mapping on . So by the Banach contraction mapping principle, we draw a conclusion that there exists a unique fixed point for in . It is clear that is a square-mean asymptotically almost automorphic mild solution of (1). The proof is complete.

The next result is proved using the similar steps as in the proof of the previous result, so we omit the details.

Theorem 22. *Assume that (H1)–(H3) hold. If and for all and , then for every , there exists a unique square-mean asymptotically almost automorphic mild solution of the problem (1) on such that .*

Theorem 23. *Suppose that assumptions (H2), (H3), and (H5) hold. If and for all and , then for every , there exists a unique square-mean asymptotically almost automorphic mild solution of the problem (1) on such that .*

*Proof. *Consider the nonlinear operator given by

First we prove that maps into itself. Given , from the properties of , and , we infer that is well defined and continuous. Since is bounded, we can choose a bounded subset of such that for all . It follows from conditions (H2) and (H3) that both and are uniformly continuous on the bounded subset uniformly for . Moreover, from Lemmas 17 and 20 and taking into account (H5), it follows that .

Next we prove that is a contraction mapping from into itself. Note that we have already proved . Moreover, for any ; and , we have
Therefore
That is, is a contraction mapping on . By the Banach contraction mapping principle, has a unique fixed point . It is clear that is a square-mean asymptotically almost automorphic mild solution of (1). The proof is then complete.

#### 4. An Example

In this section, we apply the results obtained previously to investigate the existence of square-mean asymptotically almost automorphic mild solutions for the following partial stochastic fractional differential system: where , are continuous functions, and is a fixed constant.

Let with the norm and be the operator defined by domain . It is well known that is the infinitesimal generator of an analytic semigroup on . Furthermore, is sectorial of type .

In the sequel, we assume Now, we can define the functions by which permits to transform the system (36) into the abstract system (1). Moreover, it is not difficult to see that are continuous and Lipschitz in the second variable with Lipschitz constants and , respectively.

The next result is a consequence of Theorem 22.

Theorem 24. *Under the previous assumptions, (36) has a unique mild solution whenever and are small enough.*

#### Acknowledgments

The first author was supported by Research Fund for Young Teachers of Sanming University (B2011071Q). The second author was supported by NSF of China (11361032), Program for New Century Excellent Talents in University (NCET-10-0022), and NSF of Gansu Province of China (1107RJZA091). And the third author was partially supported by Ministerio de Economia y Competitividad (Spain), project MTM2010-15314, and cofinanced by the European Community fund FEDER.

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